"In this sensitive and important historical moment, we announce to our neighbours that we extend the hand of friendship and brotherhood to them," he said.
Tensions between Iran and the US and its allies have threatened to boil over since May last year when President Donald Trump unilaterally withdrew from a 2015 nuclear deal and later began reimposing sanctions in a campaign of "maximum pressure".
The tensions escalated in the wake of devastating September 14 attacks on Saudi oil installations that Washington and Riyadh have, to varying degrees, blamed on Tehran.
Following the attacks, which caused global oil prices to spike, Trump initially raised the possibility of military retaliation, tweeting that the United States was "locked and loaded".
The United States later expanded its long list of sanctions against Iran by further targeting its central bank, as Trump indicated he did not plan a military response.
US Defence Secretary Mark Esper announced on Friday that the US was sending reinforcements to Saudi Arabia at "the kingdom's request", while noting the forces would be "defensive in nature" and focused on air and missile defence.
In his speech delivered before a massive military parade on Sunday, Rouhani called on the foreign powers in the Gulf region to "stay away".
"If they're sincere, then they should not make our region the site of an arms race," he said.
"Your presence has always brought pain and misery for the region. The farther you keep yourselves from our region and our nations, the more security there will be for our region."
"We are even ready to ignore their past mistakes... since today the situation is such that the enemies of Islam and the region, meaning America and Zionism, want to abuse the rift between us.
"Our logic is the logic of a Persian Gulf whose security comes from within," said the Iranian president.
Following his speech, the Iranian armed forces paraded their latest equipment, including tanks, missiles and armoured vehicles as Rouhani and top military commanders saluted them.
Rouhani is expected to travel to New York on Monday, a day before general debate kicks off at the United Nations General Assembly.
Foreign Minister Mohammad Javad Zarif, who is subject to US sanctions, has already arrived in New York ahead of the annual gathering.
Tensions have flared in the Gulf since May, when Iran began reducing its commitments to the nuclear deal and the US said it was sending forces to waters near the Islamic republic in response to "indications of a credible threat" from its forces.
The US deployed an aircraft carrier strike group and a bomber task force to the sensitive waterways, before sending B-52 bombers, an amphibious assault ship and Patriot missile battery.
Following the deployments, commercial ships were mysteriously attacked, drones downed and oil tankers seized in Gulf waters.
Trump in June authorised a military strike after Iran shot down a US drone, only to call off the retaliation at the last moment.
The regional crisis deepened with September 14 attacks on Saudi energy giant Aramco's Abqaiq processing plant and Khurais oilfield, which halved the kingdom's oil output.
Yemen's Iran-backed Huthi rebels claimed responsibility, but the US says it has concluded the attacks involved cruise missiles from Iran and amounted to "an act of war".
Saudi Arabia has warned of "appropriate measures" once the source of the strikes on its oil facilities is confirmed by a UN investigation it has sought.
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